What is our primary focus? Teaching Mathematics – therefore, where is the curriculum & assessment coming from. Our job should not be to create curriculum & assessment. So, where should we find these great items?

How are assessment questions aligned with instructional practice?

Characteristics of high quality assessment

Justification / explanations

Multiple strategies

Models can be used to support

Reasoning / critiquing

Fair

Aligned to standards

Limits complexity of language

DOK – what is the cognitive complexity?

Content is assessed @ DOK 1 & 2

Problem Solving – DOK 2 & 3

Communicating & Reasoning – DOK 2 & 3 (with some 4)

Modeling & Data Analysis – DOK 1-4

DOK does not equal level of achievement of student.

DOK Levels (What kind of thinking is needed to respond?)

Recall & Reproduction

Basic Skills & Concepts

Mental processing beyond recall is necessary.

Strategic Thinking & Reasoning

Extended Thinking

Excel vs Exceed – does a shift to “excel” have more meaning

Evidence of complete understanding

Evidence of reasonable understanding

Evidence of inadequate understanding

No Evidence

Rigor – the pursuit of

conceptual understanding

procedural skill & fluency

application

with equal intensity

Standards Based drivers

What should my students be able to do?

How will we know when my students are successful?

What will I do if they “got it”?

What will I do if they did not “get it”?

Assessment –

something that we do with (not to) a student.

integrated with the learning.

DOK level of instruction should be above the level of assessment

Curriculum

What you are teaching – the standards

When you are teaching – scope & sequence

How you are teaching – teacher instruction

Students must benefit from formative assessment.

Comparing Tasks – how do we improve existing tasks / assessments ?

Justification & the Frayer Model – how do the mathematics and model justify each other?

more disconnected thoughts from the MSIS at the Shanghai American School

Opening: Formative Assessment is challenging to implement because it necessitates us to adjust our teaching. It’s so much easier to believe that students walked out of the door understanding what happened than to find out that they didn’t and need to adjust.

Why MSIS – to help shift mathematical practices in schools. How do we return and improve mathematical instruction at our schools?

Start with the Problem! How else will students begin to realize what they know, what methods are efficient / inefficient, and how these problems relate.

Algebra – create the expressions from scenarios. Relate independent and dependent variables. How can a simple scenario be reworked to focus sometimes on the dependent and other times on the independent?

6.EE.2 Write, read and evaluate expressions in which letters stand for numbers.

6.EE.3 Write expressions that record operations with numbers and with letters standing for numbers (i.e. Express the calc. “Subtract y from 5 as 5-y”)

3a – Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity (i.e. 2(8 + 7) as a product of two factors; (8 + 7) as a single entity and a sum of tw terms.

3b – Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole number exponents, in the conventional order t when there are no parentheses to specify a particular order (Order of Operations)

7.EE.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related (i.e. a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05”

3c – Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2+x) to produce 6 + 3x;

6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

7.EE.3 Solve multi-step real life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions and decimals)

Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies (i.e. If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary, or $2.50 for a new salary of $27.50)

6.EE.8 Write an inequality of the form x>c or x<c to represent a constraint or condition in a problem. Recognize that inequalities of the form x>c or x<c have infinitely many solutions; represent solutions of such inequalities on number line diagrams

6.EE.9 Use variables to represent two quantities in a problem that change in relationship to one another; write an equation to express one quantity (dependent variable in terms of independent variable)

Analyze the relationship between the dep. and ind. variables using graphs and tables and relate these to equation. (example – in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d=65t to represent the relationship between distance and time.)

7.EE.4 Use variables to represent quantities in a real-world or mathematical problem and construct simple equations and inequalities to solve problems by reasoning about the quantities

a) Solve word problems leading to the equations of the form px + q = r and p(x+q)=r where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used. (ex. the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?)

b) Solve word problems leading to inequalities of the form px + q > r or px + q <r. Graph the solution set of the inequality and interpret it in the context of the problem. (ex As a salesperson, you are paid $50 / week plus $3 per sale. This week you want your paty to be at least $100. Write an inequality for the number of sales you need to make and describe the solutions.

Last weekend, I joined several math teachers on a trip to Shanghai. We began a series of “institutes” that will span the next two years. The Math Specialist in International Schools (MSIS) seems to be a great opportunity to talk with other math teachers, think about my practice and find ways to improve. Presented by Erma Anderson and Steve Leinwand, the weekend focused on developing number sense and a progression through the Common Core standards.

On the first day, we spent most of the time discussing systems in the early primary years. With two girls in preK-3, my attention was grabbed as the importance of thinking more in terms of age instead of grade was repeated. I can only hope that the journeys of my daughters will be full of active exploration, manipulating models and discourse.

Rich mathematical tasks quickly became the focus of the institute. Students must engage with mathematics by grappling with problems, developing solutions, revising strategies and talking about their thinking. Have I been giving enough space for all of this to happen? My personal bank of resources has grown over the years but I soon realized how small shifts in my presentation of tasks can give massive dividends in the end.

Ease into the problem. “What do you notice?”

Build excitement. “What do you wonder?”

Turn the keys over. “What is the question?”

Invest the activated minds. “What is a value that you know is too high?”

Build skills of estimation. “What is a value that you know is too low?”

But don’t simply ask and be satisfied. Question. Have students explain their thinking. Over and over and over. This is a great way to review concepts and flush out activities. “Excuse me. You said that the object is 3D. What do you mean by 3D?”

“I’m not sure that I understand. Can you tell me more about the dimensions? The units? the…”

Finally, primed minds are released to tear the problem apart. But continue to push. Convince me. Show me. (Yep, that means providing multiple representations.) Explain.

The take home for me was to slow down and question. I need to do a better job to anticipate the reaction of students and be ready for targeted follow-up questions. How does this look? After more than a decade of teaching, I’m ready to consider creating a presentation a la .ppt. So far, I’ve done more hopping into the rich task through a video, image or description but I think that I’ve lost many opportunities in the set-up. I need to slow things down, expect communicate and question more.

The opposite bookend is equally important: presentation of the solution. My work is cut-out for me. I need to expand upon my own strategies of math talks related to solutions and student thoughts. The work of students should be more directed in the deepening of understanding and demonstration of new strategies.

Typically not one to slowly invest, my first class after the weekend focused on a rich task nestled inside a presentation. I posed probing questions to responses that I often would not at and move on. We took more time setting up the problem but the conservations were rich. When it was time to start work, the class erupted in a flurry of activity. The debrief was also richer as again I focused on asking for more explanations and stopped accepting values/comments without being convinced by the student.

A final thought: How can I shift more of my class to better scaffolded, rich tasks?

On the last day of the institute, we sat down in grade-level groups. This was the first time that I had worked with this group of teachers. In fact, introductions were the first order of business. We were then asked to put-it-into-practice: create a lesson. In a relatively short amount of time, we identified a rich task aligned with our goal and designed a scaffolded plan to work with students. What if more collaboration / professional development times were spent this way…